Suppose we have 3 people standing around and they want to meet. Normally that would be easy but they have quite a lot of restrictions.
- The only movement thay can make is to move towards the centroid of the triangle they form.
- Only one person can move at a time.
- A person follows this pattern of thought: 1) Firstly they look around and see where the others are(i.e. calculate the triangle and by extension the centroid). They then move a random distance towards the centroid they calculated.
Here's the catch though, between looking and moving another person can also look and move and if he does move then it will "mess up" the calculations of the person he interrupted resulting in a different triangle's centroid than the one the person who will resume will move towards to.
My questions is if they will ever meet. My instict says yes since they will always move closer to each other, as in even if they move towards the centroid thay calculated before it changed they will get closer and at some point gather, but i do not know how to prove it.
How close do people have to be to meet? 3 bodies cannot occupy the same space. So to have a successful meeting they just have to be in the same neighborhood.
And as each person moves, the average / maximum distance between the 3 gets smaller. With some number of iterations they will be arbitrarily close to one another.